Encipherment of digital sequences by reversible transposition methods

ABSTRACT

Methods for transposing elements of a sequence according to a rule, wherein the rule is derived from pseudo-noise or pseudo-noise like binary and non-binary sequences are disclosed. Sequences of transposed symbols can be recovered by applying a reversing rule. Sets of orthogonal hopping and transposition rules are created by applying transposition rules upon themselves. Sets of orthogonal hopping and transposition rules are also created from binary and non-binary Gold sequences.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication Ser. No: 60/720,655, filed Sep. 26, 2005, which isincorporated herein by reference.

BACKGROUND OF THE INVENTION

This invention relates to the encipherment of binary and non-binarydigital sequences such as used in communications by reversibletransposition methods and the decipherment of sequences enciphered byreversible transposition methods. More specifically it relates toapplying methods using recoded binary and non-binary pseudo-noisesequences generated by LFSR based sequence generators and other methodsthat will generate reversible sequences.

Sequences comprised of digital elements have known applications incommunications and other applications. In general binary pseudo-noise orPN-sequences are used. Application of non-binary sequences is alsopossible. Linear feedback shift register (LFSR) circuits or methods areoften used for the generation and detection of sequences. LFSR circuitswith p register elements can generate (n^(p)−1) length unique n-valuedsequences which are called maximum-length sequences. It is oftendesirable to encrypt digital data for transmission, or storage on adata-storage medium such as optical disks or as an embedded message forwatermarking applications.

Substitution ciphers are known, wherein according to some rules onesymbol or series of symbols is replaced by another. Another enciphermentmethod is transposition wherein in a series of symbols the order of thesymbols is changed according to a rule or set of rules.

While transposition encipherment can be used for security reasons, itcan also be used to randomize a process in a recoverable way. One suchapplication is the creation of sequences for application in frequencyhopping in telecommunications. In many cases binary LFSR basedpseudo-random sequences are used as a number generator to create hoppingrules. Orthogonality of the sequences is important so that each user ina hop is assigned a unique frequency slot. Another application is intime-hopping applications. Herein each user is assigned a uniquetime-slot, so that pulses of different users do not collide. Non-binarypseudo-random sequences have statistical advantages over the generallyused binary sequences. It is often useful to have a local method togenerate the transposition rule as well as the rule to recover thetransposed sequence. Also the ability to select from a large number ofpossible encipherment rules is advantageous.

Accordingly, new methods for symbol transposition in a pseudo-randomlike fashion are required.

SUMMARY OF THE INVENTION

In view of the more limited possibilities of the prior art inenciphering binary and non-binary digital sequences by transposition,the current invention provides methods and apparatus for the rules ofencipherment by transposition of digital sequences and the deciphermentof the encrypted sequences.

The general purpose of the present invention, which will be describedsubsequently in greater detail, is to provide novel methods andapparatus which can be applied in the encipherment by transpositionusing digital sequences with pseudo-noise or pseudo-noise likeproperties and the decipherment of the encrypted sequences. Sequencesare made of series of symbols with an assigned position relative to anassumed or assigned origin or anchor point. The individual symbols andtheir order in a sequence may represent an electrical or optical signal.The position of a symbol may represent a physical order, a time slot, afrequency, a color or any other phenomenon or concept that can berepresented as a position.

Before explaining at least one embodiment of the invention in detail, itis to be understood that the invention is not limited in its applicationto the details of construction and to the arrangements of the componentsset forth in the following description or illustrated in the drawings.The invention is capable of other embodiments and of being practiced andcarried out in various ways. Also, it is to be understood that thephraseology and terminology employed herein are for the purpose of thedescription and should not be regarded as limiting.

Binary in the context of this application means 2-valued. Multi-valuedand n-valued in the context of this invention mean an integer greaterthan 2.

It is one aspect of the present invention to provide new methods totranspose symbols in a sequence of digital symbols in a recoverable orreversible manner.

It is another aspect of the present invention to enable detection oftransposed sequences by knowing the transposition rule.

It is a further aspect of the present invention to provide a method forcreating transposition rules based on pseudo-random binary andnon-binary sequences.

It is another aspect of the present invention to provide a method forcreating transposition rules based on pseudo-random sequences with noforbidden word.

It is a further aspect of the present invention to provide a method ofcreating plurality of orthogonal hopping rules by repeated applicationof a transposition rule.

It is another aspect of the present invention to provide a method forcreating a plurality of orthogonal hopping rules based on Goldsequences.

It is a further aspect of the present invention to provide a systemwhich implements the methods provided as different aspects of thepresent invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Various other objects, features and attendant advantages of the presentinvention will become fully appreciated as the same becomes betterunderstood when considered in conjunction with the accompanyingdrawings, and wherein:

FIG. 1 is a block diagram of a binary LFSR based sequence generator.

FIG. 2 shows a correlation graph.

FIG. 3 is a diagram of a transposition rule.

FIG. 4 is a diagram of another transposition rule.

FIG. 5 is a diagram of an LFSR based sequence generator.

FIG. 6 shows a correlation graph.

FIG. 7 shows another correlation graph.

FIG. 8 shows a cross-correlation graph.

FIG. 9 is the diagram of an LFSR based sequence generator

FIG. 10 is the auto-correlation graph of a transposed 26 symbolsequence.

FIG. 11 is the cross-correlation graph of an original m-sequence withits transposed sequence.

FIG. 12 is the auto-correlation graph of a transposed ternary m-sequencewhich is transposed again with the ‘modulo-n+1’ rule.

FIG. 13 is the cross-correlation graph of the transposed ternarym-sequence with the sequence created by ‘modulo-n+1’ transposition ofthis sequence.

FIG. 14 is the cross-correlation graph of the original ternarym-sequence with the ‘modulo-n+1’ transposition of the transposed ternarym-sequence.

FIG. 15 shows the combined auto-correlation graph of a sequence combinedwith the cross-correlation graph of this sequence with another sequence.

FIG. 16 shows the auto-correlation graph of a 16 element 4-valuedsequence

FIG. 17 shows the auto-correlation graph of a transposed sequence

FIG. 18 shows the cross-correlation graph of a sequence with itstransposed sequence.

FIG. 19 shows a pulse train diagram for time hopping

FIG. 20 shows another pulse train diagram

FIG. 21 is a diagram of a transposition system in accordance with anaspect of the present invention.

FIG. 22 is a diagram of a transposition reversing system in accordancewith an aspect of the present invention.

FIG. 23 is a diagram of a frequency hopping system in accordance with anaspect of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The Related Art

There are different ways to transpose symbols in a sequence or a wordcomprised of n symbols. In general one can create transpositions byselecting one of the possible reversible permutations of a word of nsymbols. A symbol may be represented by a single element or a number ofelements. A symbol as one aspect of the present invention will beassumed to being able to be represented by a single element, keeping inmind that it can in actuality be represented by a plurality of elements.A word or sequence of n symbols has n!=1*2*3*4* . . . *(n−1)*n possiblepermutations. A sequence of n symbols may comprise p different symbolswherein p<n. In that case some permutations are of course identical.

A transposition rule may be created by a pseudo-random number generator.A reversible transposition rule for the transposition of n symbols maybe considered an n valued reversible inverter.

Pseudo-random sequences as transposition rules.

The inventor describes ‘word-based’ methods for generating pseudo-randomand pseudo-random like binary and non-binary sequences in U.S.Provisional Patent Application Ser. No. 60/695,317 filed on Jun. 30,2005 entitled THE CREATION AND DETECTION OF BINARY AND NON-BINARYPSEUDO-NOISE SEQUENCES NOT USING LFSR CIRCUITS and in U.S. patentapplication Ser. No. 11/427,498 filed on Jun. 29, 2006 entitled THECREATION AND DETECTION OF BINARY AND NON-BINARY PSEUDO-NOISE SEQUENCESNOT USING LFSR CIRCUITS which are hereby incorporated by referenceherein in its entirety. One aspect of that invention demonstrates thatan n-valued pseudo-noise or maximum length sequence which can begenerated by an LFSR circuit with a shift register with p memoryelements of a length of (n^(P)−1) symbols, can also be generated by amethod combining in a prescribed fashion (n^(p)−1) unique words of pn-valued symbols and taking from each word one symbol to create thesequence. The order of the symbols in the sequence is determined by theorder of the words.

The words are identical to the consecutive states of the shift register.Each word is unique until it repeats itself again. In an LFSR the repeatis after (n^(p)−1) states. The words may be assumed to represent adecimal number. Within the (n^(p)−1) cycles each word and itsrepresentative decimal number in a pseudo-random sequence is unique. Onemay consider the order of the decimal numbers then as a reversibleshuffling rule or a transposition rule.

For instance a binary LFSR pseudo-noise generator with a 3 element shiftregister can generate a pseudo-noise sequence of length (2³−1)=7elements. The following table shows how the sequence can be generated byusing overlapping words of 3 bits. As an illustrative example thegenerated sequence is [0 0 1 1 1 0 1] created from the first bit of eachword. TABLE 1 se- s3 s2 s1 dec1 dec2 quence out1 0 0 1 1 4 0 out3 0 1 13 6 0 out7 1 1 1 7 7 1 out6 1 1 0 6 3 1 out5 1 0 1 5 5 1 out2 0 1 0 2 20 out4 1 0 0 4 1 1

FIG. 1 shows the diagram of the LFSR based circuit that will generatethe binary pseudo-noise sequence. The initial (or seed) state of theLFSR is [0 0 1]. The forbidden state in this configuration is [0 0 0].FIG. 2 shows the auto-correlation graph of the generated sequence.

The Table 1. shows a column ‘dec1’, which is the radix-10 value of the 3bit word, with the most significant bit being the first bit of the word.The column ‘dec2’ in the table shows the radix-10 value of the 3-bitword with the last bit being the most significant bit of each 3-bitword.

The columns in the Table 1. under ‘dec1’ and ‘dec2’ can be interpretedas a rule for a transposition of symbols in a sequence. Thistransposition may be considered orthogonal in the sense that each symbolwill be transposed to a unique new position, in such a way that thetransposition can be reversed and no position will be shared by twosymbols.

Clarification about the transposition rules will be provided next. Firstof all it should be clear what the transposition rule actually means.Two possible different ways to apply a transposition rule are providedin the following Table 2 and Table 3. TABLE 2 sequence Origin dec1trans_to*1 seq_res1 trans_to*2 seq_res2 Inverse 0 1 1 1 0 1 0 1 0 2 3 60 4 1 6 1 3 7 2 0 6 0 2 1 4 6 7 1 3 1 7 1 5 5 5 1 5 1 5 0 6 2 4 1 7 1 41 7 4 3 1 2 0 3

Table 2 and Table 3 show 2 different interpretations of thetransposition rules. In Table 2 the transposition column under ‘dec1’means that a symbol on a position as stated in the column under Originis being transposed to the position as stated in the column under ‘dec1’in Table 2. So the symbol in the original position 1 is being transposedto position 1. The symbol in position 2 is transposed to position 3; thesymbol in position 3 is being transposed to position 7; etc. The resultof that transposition is shown in Table 2 in the column under‘trans_to*1’. Or in other words: the first position of the sequence as aresult of the transposition has a symbol that originally was in thefirst position of the un-transposed sequence. The second symbol in thetransposed sequence is the symbol originally in the 6th position; thethird symbol in the transposed sequence is the second symbol in theoriginal sequence; etc.

In order to confirm the rule it is executed an additional time on thecolumn under ‘trans_to*1’ and the result of the transposition is shownin the column under ‘trans_to*2’. The numbers in the column indicate theoriginal position of the symbols. The actual transposed sequence isshown in the column in Table 2 ‘seq_res2’. The column under Inverseshows the inverse transposition to the rule of ‘dec1’. The InverseTransposition applied to the result in column ‘trans_to*1’ will createthe original sequence. Applying the Inverse Transposition twice to theresult under ‘trans_to*2’ will also recreate the original sequence. Thetransposition by ‘trans_to’ rule and its reverse is shown graphically inFIG. 3. TABLE 3 sequence Origin dec1 trans_from*1 seq_res1 trans_from*2seq_res2 Inverse 0 1 1 1 0 1 0 1 0 2 3 3 1 7 1 6 1 3 7 7 1 4 1 2 1 4 6 60 2 0 7 1 5 5 5 1 5 1 5 0 6 2 2 0 3 1 4 1 7 4 4 1 6 0 3

Table 3 shows the transpositions by the ‘transpose from’ rule.

The ‘transpose from’ interpretation of the rule means that the rule asdisplayed in Table 3 in the column under ‘dec1’ means that the symbol ina certain position in the transposed sequence comes from the positionindicated by the number on that position. The result is shown in Table 3in the column under ‘trans_from*1’. The symbol in position 1 of thetransposed sequence is coming from position 1 of the original sequence.The symbol in position 2 of the transposed sequence comes from position3 of the original sequence. Etc, etc. The actual transposed binarysequence is shown in Table 3 in the column under ‘seq_res1’.

One can apply the transposition rule on the transposed sequence whichwill result in the transposition as shown in Table 3 in the column under‘trans_from*2’. The actual twice transposed sequence is shown in Table 3in the column under ‘seq_res2’.

The Inverse Transposition rule to reverse this example of a ‘TransposeFrom’ rule is shown in Table 3 in the column under ‘Inverse’. Theinversion rules for both ‘from’ and ‘to’ rules look identical, but areof course applied differently. In fact the ‘transpose to’ rule is theinverse of the ‘transpose from’ rule. The transposition by ‘trans_from’rule and its inverse is shown in FIG. 4.

Pseudo-code

The ‘to’ and ‘from’ transposition rules can be more easily explained incomputer program pseudo-code. The following Table 4 shows four columns,each with 7 elements. TABLE 4 origin rule result_to result_from 1 1 1 12 3 6 3 3 7 2 7 4 6 7 6 5 5 5 5 6 2 4 2 7 4 3 4

The ‘to’ transposition rule can be written as: FOR i=1:7   INDEX=RULE(i)  RESULT_TO(INDEX)=ORIGIN(i) NEXT

The transposition_to rule as provided in the table can be applied to asequence of 7 symbols seq=[a b c d e f g]. Applying the rule will leadto [a f b g e d c]. It should be clear that the symbols ‘a’, etc areselected as different characters to differentiate between them. Thevalue of a symbol is to be determined separately from the transpositionrule and can be any n-valued symbol.

The ‘from’ transposition rule can be written as: FOR i=1:7  INDEX=RULE(i)   RESULT_FROM(i)=ORIGIN(INDEX) END

Applying the transposition_from rule from Table 4 to [a b c d e f g]will provide [a c g f e b d].

For simplicity reasons the ‘transpose from’ method will be used forillustrative purposes to describe the present invention. The reason forthat is that the rule provides the index of the transposition. The“transposition to” requires an intermediate step to display the index ofthe transposed sequence. This is not fundamental to the method, but maybe confusing. It should be clear that the ‘transpose to’ method can alsobe used in the provided examples.

The Forbidden Initial State

LFSR based sequence generators have an initial state of the shiftregister that is known as the ‘forbidden’ state. For instance when theLFSR applies only binary XOR functions, the ‘forbidden’ state of theshift register is all 0. In general a ‘forbidden’ state of a shiftregister does not create any change. This means that the shift registerfeedbacks into the feedback functions create a new input to the shiftregister, followed by a clock-pulse with a shift of the content of theregisters in such a way that the new content of the shift register isidentical to the previous content. Because the shift register does notchange its content, the output of the circuit will be a sequence ofidentical symbols (in this case 0s), which in many cases is notdesirable.

One can use the binary EQUAL function as feedback function. In that casethe all 1 content of the shift register will be the forbidden state. Thesame phenomenon of ‘forbidden’ states will occur in higher value orn-valued LFSR based sequence generators. The ‘forbidden’ state willdepend on the applied n-valued functions in the feedback path.

The significance of the ‘forbidden’ state is that it creates an LFSRword that does not occur in an allowed sequence generator. That meansthat this word does not occur in an LFSR PN generator basedtransposition rule. In the binary case using an LFSR using only XORfunctions it means that the word comprising all 0s does not occur. Thisis not a problem in a transposition wherein a position 0 does not occur.However in the case where all Is cannot occur (using only the EQUALfunction) it means that all 0s can occur and all Is cannot. This againmeans that the transposition rule includes a position 0.

As an illustrative example of the effect of the selected feedbackfunction the LFSR circuit of which a diagram is shown in FIG. 5 will beused. The circuit 500 in FIG. 5 is a binary LFSR based pseudo-noisesequence generator. With a 4-element shift register the circuit 500generates a PN sequence of length 15. The circuit has a single tap 506with a binary function 502.

When the function of device 502 is the binary XOR function and theinitial state of the shift register is [0 1 1 0] then the generated15-bits PN sequence is: [0 1 1 0 0 1 0 0 0 1 1 1 1 0 1]. Thetransposition rule (equivalent with ‘dec1’ in the previous example)formed from overlapping 4 bits words is: [6 12 9 2 4 8 1 3 7 15 14 13 105 11].

When the function of device 502 is the binary EQUAL function and theinitial state of the shift register is [0 1 1 0] then the generated15-bits PN sequence is: [0 1 1 0 1 1 1 0 0 0 0 1 0 1 0]. Thetransposition rule (equivalent again with ‘dec1’ in the previousexample) is: [6 13 11 7 14 12 8 0 1 2 5 10 4 9 3]. The ‘forbidden’ statein this case is [1 1 1 1] which is equivalent with ‘dec1’ position 15and cannot occur in this example. The initial state [0 0 0 0] is validand will occur and is equivalent with ‘dec1’ position 0. This positionwill occur in ‘dec1’. In general positions of symbols in a sequence arerepresented by using 1 as the start position. In order to make ‘dec1’ inthis case a usable transposition rule one should add a 1 to allpositions. In that case the transposition rule becomes: [7 14 12 8 15 139 1 2 3 6 11 5 10 4].

The Rule for Reversing the Transposition

The ‘transpose from’ rule is used to illustrate one aspect of thepresent invention. It should be clear that the ‘transpose to’ rule willin fact reverse the transposition. If one transposes a sequence forinstance twice then the resulting sequences has to be reverse transposedtwice to recreate the original sequence. One can also recreate orinverse the transposition by applying the transposition rule apre-determined number of times. This is demonstrated in the followingTable 5. TABLE 5 orig- start [0 1 1] start [0 1 0] inal 1 2 3 4 5 6 7 12 3 4 5 6 7 t1 3 7 6 5 2 4 1 2 4 1 3 7 6 5 t2 6 1 4 2 7 5 3 4 3 2 1 5 67 t3 4 3 5 7 1 2 6 3 1 4 2 7 6 5 t4 5 6 2 1 3 7 4 1 2 3 4 5 6 7 t5 2 4 73 6 1 5 t6 7 5 1 6 4 3 2 t7 1 2 3 4 5 6 7

The left side of the table shows the transposition rule generated by thesequence generator of FIG. 1 with initial shift register content [0 11]. This table shows that after 7 transpositions the original situationhas been recreated. The right side of the table shows the transpositionrule generated by the same circuit but with initial condition [0 1 0].It takes 4 transpositions to recreate the original situation.

This shows that a new transposition rule can be created by using thesame transposition rule more than once. However one should be carefulnot to use the transposition rule too many times and thus recreate theoriginal. Also changing the initial state of the LFSR will create adifferent transposition rule.

Another way to change a transposition rule of n positions is bydetermining the modulo-n residue of the rule and then adding a 1.

One can interpret the generated LFSR words which will form thetransposition rule in reverse. That is: instead of interpreting [0 0 1]as decimal 1 one can read this a decimal 4. This will change thetransposition rule for a specific initial value of the LFSR, however itdoes not change the principle. In that case a binary LFSR with ann-element shift register will also generate (2^(n)−1) different numbersranging from either 1 to (2²−1) or from 0 to (2^(n)−2).

The Effects of Transposition

One way to show the effects of transposition is to submit a known binarypseudo-noise sequence to a transposition rule. The quality oftransposition can be demonstrated by the auto-correlation andcross-correlation graphs. The sequence that will be transposed isseqbin_(—)31=[1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 11 0]. This sequence is a pseudo-noise sequence of which theauto-correlation graph is shown in FIG. 6. This sequence will betransposed using a rule generated by a 5 element binary shift registerwith initial content [0 1 1 0 0].

The transposition rule is:

-   [1 2 6 3 1 16 8 20 26 13 22 11 21 10 5 2 17 24 28 14 23 27 29 30 31    15 7 19 9 4 18 25].

Transposing sequence seqbin_(—)31 with this rule creates:

-   seqbin_(—)31_trans=[0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0    0 1 1 0 1 1 1]. The auto-correlation graph of seqbin_(—)31trans is    shown in FIG. 7. The cross-correlation graph of the sequences    seqbin_(—)31 and seqbin_(—)31_trans is shown in FIG. 8.

It should be clear that correlation between the original sequence andthe transposed sequence will not be helpful in detecting the sequence.

The inverse rule of a transposition rule can be expressed in thefollowing pseudo-code: FOR i=1: n IND = RULE (i) INVERSE(IND) = i ENDThe vector RULE is the transposition rule vector. The vector INVERSE isthe inverting rule. This approach applied to the above transpositionrule for a 31 symbol sequence will generate INVERSE=[4 15 3 29 14 2 26 628 13 11 1 9 19 25 5 16 30 27 7 12 10 20 17 31 8 21 18 22 23 24].Applying rule INVERSE to sequence ‘seqbin_(—)31_trans’ will recreatesequence ‘seqbin_(—)31’.

It should be clear that absolute synchronization of sequences isrequired in applying the transposition rules.

The Word Method

One problem with using LFSR related methods for generating TranspositionRules is that it can only generate Rules of length (n^(p)−1) or in thebinary case (2^(p)−1) when the LFSR has shift registers with p elements.The inventor has demonstrated in the earlier cited Provisional PatentApplication that one can use the so called ‘word’ method to extend thesequence with one symbol or the number of ‘words’ with one more. In thatcase the number of words is a multiple of n. Or in the binary case amultiple of 2. This is advantageous in some applications. An aspect ofthe present invention is to generate a transposition rule that cantranspose sequences with an even number of symbols.

The invention of the ‘word’ method to generate binary sequences of evenlength is described in detail in U.S. Provisional Patent Application No.60/695,317 filed on Jun. 30, 2005 entitled CREATION AND DETECTION OFBINARY AND NON_BINARY PSEUDO-NOISE SEQUENCES NOT USING LFSR CIRCUITS.One example will be repeated here for illustrative purposes only.

In this example unique 4-bits binary words are used in such a way thatthe last three bits of a word are identical to the first three bits of anext word. For instance the first bit of each word will also be a bit inthe sequence. There are of course 16 4-bit words. There are many ways tocreate partial solutions wherein less than 16 words are used. Well knownsolutions are pseudo-random sequences, formed in such a way that [0 0 00] or [1 1 1 1] are not used. In that case the resulting sequence willhave a length of 15 bits.

There is at least one way to create a solution of 16 words, which isshown in the following table. b1 b2 b3 b4 dec word1 1 1 1 1 15 word2 1 11 0 14 word3 1 1 0 1 13 word4 1 0 1 1 11 word5 0 1 1 0 6 word6 1 1 0 012 word7 1 0 0 1 9 word8 0 0 1 0 2 word9 0 1 0 1 5 word10 1 0 1 0 10word11 0 1 0 0 4 word12 1 0 0 0 8 word13 0 0 0 0 0 word14 0 0 0 1 1word15 0 0 1 1 3 word16 0 1 1 1 7 seq16 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0The 4-bit words are shown as decimal numbers in the table in the columnunder ‘dec’. This column contains 16 numbers, including 0. It may beconsidered as a sequence generator with no forbidden state. In order tomake the column a valid transposition rule, all numbers have to beincreased with 1. The transposition rule is then: trans_(—)16=[16 15 1412 7 13 10 3 6 11 5 9 1 2 4 8].

The inverse rule is: inv_trans_(—)16=[13 14 8 15 11 9 5 16 12 7 10 4 6 32 1].

Superimposing Different Transposition Rules

It should be clear that it is possible to ‘superimpose’ differenttransposition rules on a sequence. Additional security may be obtainedby using transposition rules of different lengths. For instance asequence of 32 symbols can be broken up in two contiguous sequences oflength 7 and 25. As will be shown as another aspect of the presentinvention one can generate transposition rules of length 25 by usingmulti-valued methods. A transposition rule of length 7 can be generatedby for instance a 3-element LFSR. And a 32 symbol transposition rule canbe generated by a 5-bit word method.

One way to create a super-imposed method is to transpose the first 7symbols of a 32 symbol sequence with a length 7 rule. Then transpose theremaining 25 symbols with a 25 length rule. And next transpose thecombined transposed sequences with a 32 length rule.

Another way to create a super-imposed transposition method is to firsttranspose the 32 symbol sequence with a length 32 rule and then executethe 7 symbol and the 25 symbol transposition. All transpositionsaccording to the present invention are reversible if the transposedsequence remains synchronized with the original ‘not transposed’sequence. A sequence that was enciphered by using superimposed rules canbe recovered in its original form by applying the inverses of each rulein reverse order of their application.

Ternary Transposition Rules

Another aspect of the present invention is the creation of transpositionrules based on generating ternary or 3-valued pseudo-noise orpseudo-noise like sequences.

The inventor has shown in U.S. Non-Provisional patent application Ser.No. 10/935,960, filed on Sep. 8, 2004, entitled TERNARY AND MULTI-VALUEDIGITAL SCRAMBLERS, DESCRAMBLERS AND SEQUENCE GENERATORS which isincorporated herein by reference in its entirety, how one can generateternary or 3-valued maximum length sequences with LFSR methods.Consequently, as was shown in U.S. Provisional Patent Application No.60/695,317 filed on Jun. 30, 2005 entitled CREATION AND DETECTION OFBINARY AND NON_BINARY PSEUDO-NOISE SEQUENCES NOT USING LFSR CIRCUITS,all consecutive ‘words’ formed by the content of the ternary shiftregister will be unique and non-repeating for (3^(p)−1) words. Thefactor p is the length of the shift register.

The following illustrative example of a transposition rule base on aternary LFSR generated m-sequence is provided. A ternary LFSR basedsequence generator is shown in FIG. 9. The shift register comprises 3elements of which each can hold an element with one of three states. Theinitial state of elements 903, 904 and 905 is [0 1 2]. The truth tableof the applied ternary logic function in device 902 is ‘ter1’ and isshown in the following table: ter1 0 1 2 0 0 2 1 1 1 0 2 2 2 1 0The truth table is non-commutative. The columns of the truth table of‘ter1’ are determined by the output signals provided by the shiftregister element 905. The transposition rule is formed by the decimalvalue of each word formed by the content of the shift register at eachdifferent and consecutive state of the LFSR.

The generated sequence by the circuit of FIG. 9 is seq3=[2 2 2 0 0 1 0 12 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0] and has a length of 26=(27−1)symbols. The rule formed by the consecutive states of the LFSR is:Rule3=[5 19 24 26 8 2 9 3 10 21 16 14 22 7 11 12 13 4 1 18 6 20 15 23 2517]. The inverting rule is [19 6 8 18 1 21 14 5 7 9 15 16 17 12 23 11 2620 2 22 10 13 24 3 25 4].

The forbidden state of this circuit is [0 0 0]. So the decimal number 0will not occur in the transposition rule. One can create additionalrules by either changing the initial state of the LFSR or by repeatedlyapplying the rule on itself. Applying the rule once upon itself willcreate [8 1 23 17 3 19 10 24 21 6 12 7 20 9 16 14 22 26 5 4 2 18 11 1525 13].

Elsewhere, such as in U.S. Non-Provisional patent application Ser. No.10/935,960, filed on Sep. 8, 2004, entitled TERNARY AND MULTI-VALUEDIGITAL SCRAMBLERS, DESCRAMBLERS AND SEQUENCE GENERATORS which is herebyincorporated by reference herein in its entirety, the inventor has shownthat by using different ternary logic functions in LFSRs additionalternary m-sequences can be created. These additional ternary m-sequencescan in principle also be used to create orthogonal transposition rules.One potential problem can occur if the all 0 state is not the forbiddenstate. In that case the all 0 word is allowed and will be atransposition state. Because it is assumed that the sequences to betransposed and the transposed sequences start at position 1, theoccurrence of a position 0 is problematic. The following illustrativeexample will be used to demonstrate how the potential problem can beaddressed.

Assume that an LFSR as shown in FIG. 9 will be applied, with the device902 executing ternary logic function ‘ter2’ of which the truth table isshown in the following table. ter2 0 1 2 0 1 0 2 1 2 1 0 2 0 2 1

The LFSR state [1 1 1] equivalent with decimal value 13 is the forbiddenstate while [0 0 0] is allowed and will occur. Starting with [0 1 2] thefollowing 26 unique words or represented by the shift register contentwill be generated and shown in the following table. The decimalequivalent value of the ternary words is also included. word dec 0 1 2 50 0 1 1 0 0 0 0 1 0 0 9 1 1 0 12 2 1 1 22 1 2 1 16 2 1 2 23 0 2 1 7 2 02 20 2 2 0 24 0 2 2 8 1 0 2 11 2 1 0 21 2 2 1 25 2 2 2 26 1 2 2 17 1 1 214 0 1 1 4 1 0 1 10 0 1 0 3 2 0 1 19 0 2 0 6 0 0 2 2 2 0 0 18 1 2 0 15

There are several ways to make this a valid transposition rule. One wayis to replace 0 with the forbidden state (which represents 13). Thetransposition rule then becomes: [5 1 13 9 12 22 16 23 7 20 24 8 11 2125 26 17 14 4 10 3 19 6 2 18 15]. The following illustrative exampleswill show the results of several transposition rules. In the firstexample the ternary m-sequence generated by the circuit of FIG. 9 with902 realizing function ‘ter2’ will be transposed by the rule generatedby the circuit of FIG. 9 with 902 realized by ‘ter1’ and initial state[0 1 2]. The LFSR has shift register elements 903, 904 and 905; a tap906; and an output 907. The ternary m-sequence to be transposed is: [0 01 1 2 1 2 0 2 2 0 1 2 2 2 1 1 0 1 0 2 0 0 2 1 0]. The appliedtransposition rule is: Rule3=[5 19 24 26 8 2 9 3 10 21 16 14 22 7 11 1213 4 1 18 6 20 15 23 25 17] which is generated by the circuit of FIG. 9with device 902 realized by ternary function ‘ter1’. The transposedsequence is: [2 1 2 0 0 0 2 1 2 2 1 2 0 2 0 1 2 1 0 0 1 0 2 0 1 1].

The auto-correlation graph of the transposed sequence is shown in FIG.10. The cross-correlation graph of the original sequence with thetransposed sequence is shown in FIG. 11.

It is one aspect of the present invention to create a new transpositionrule from an existing rule of n positions, by determining the modulo-nresidue of all rule values and adding 1. Applying this method to theillustrative ternary example will create the rule: [6 20 25 1 9 3 10 411 22 17 15 23 8 12 13 14 5 2 19 7 21 16 24 26 18]. While it appears tobe easy to determine this rule from its source, the same is not true forthe transposed sequences. Applying the ‘modulo-n and plus-1’ rule to thesequence that was the result of the previous transposition will createthe sequence: [0 0 1 2 2 2 2 0 1 0 2 0 2 1 2 0 2 0 1 0 2 1 1 0 1]. Theauto-correlation graph of this sequence is shown in FIG. 12.

There are no obvious recognition or synchronization points between thisnewly transposed sequence and the previous one. FIG. 13 shows thecross-correlation graph of the transposed sequence with the sequencecreated by transposition the transposed sequence again with the‘modulo-n+1’ rule. There is no clear alignment between the twosequences. FIG. 14 shows the cross-correlation of the sequence createdby the ‘modulo-n+1’ rule with the original ternary m-sequence. Also inthis case there is no clear alignment.

The application of the transposition rules derived from ternary LFSRbased sequences to other LFSR generated ternary sequences is forillustrative purposes only as to make sure that the sequences as well asrules have the same number of positions and to demonstrate that apparentpositional relationships (as shown in correlation graphs) will be brokenup by the transposition rules. The transpositions of course work for anysequence of n-valued symbols. The requirement is that one can create asequence of (n−1) n-valued symbols or n^(p) n-valued symbols from wordsof p n-valued symbols in such a way that each word of p consecutivesn-valued symbols in the sequence are unique with regards to one another.This is a different way to say that the sequences should bepseudo-random.

One should make sure that the sequences to be transposed containsufficient symbols to apply the relevant transposition rules. When asequence does not contain enough symbols one may have to stuff or pad asequence with additional symbols when a transposition rule requiresadditional symbols.

Sequences of Length 3^(p)

In the cited U.S. Provisional Patent Application related to generatingsequences not using LFSR methods the inventor has shown it to bepossible to generate n-valued sequences of length n^(p) wherein wordsare used of p n-valued symbols. It is possible to arrange the words insuch a way that for instance the last (p−1) symbols of a word coincidewith the first (p−1) symbols of the next word. It is then possible touse all n^(p) words just once in creating a sequence. One can actuallyrecreate the used words from the sequence by starting to take the firstp symbols, shift one position and take again the p consecutive symbols,etc. Each word is then unique. One can create words of more than psymbols. In that case all words will be unique. However when sortedthese words (of more than p symbols) do not have to form a mainlycontiguous series of numbers, with only the forbidden word(s) missing.

It was shown in the cited patent application that the auto-correlationof these maximum length sequences can be attractive, by having onecentral high peak and much lower non-peak values. One of these sequencesof length 27 will be used as an illustrative example to generatetransposition rules. The sequence is [0 0 0 1 0 1 1 1 2 1 0 2 0 0 2 1 20 2 2 2 0 1 2 2 1 1]. This sequence of 27 ternary symbols was created bythe ‘word’ method using 27 different words of 3 symbols. This sequencecan be translated into a rule of 27 different decimal numbers: [0 1 3 104 13 14 16 21 11 6 18 2 7 23 15 20 8 26 24 19 5 17 25 22 12 9]. Becauseof the nature of the method to generate the sequence all 27 words(including all 0s) will be used. Consequently one has to add 1 to allnumbers to create a transposition rule: [1 2 4 11 5 14 15 17 22 12 7 193 8 24 16 21 9 27 25 20 6 18 26 23 13 10]. All previously mentionedmethods (inverting, shifting positions, repeated application and‘modulo-n+1’) can be applied using this rule.

For illustrative purposes the initial state of sequences in the exampleswere selected as [0 0 0]. This means that a self mapping first state ofthe transposition will be created. It should be clear that one may startwith a different state to prevent the first state to be self mapping.

Hiding One Sequence in Another Sequence

As an illustrative example it is shown what will happen when thesequence is analyzed using words of 4 elements rather than 3. Thecreated rule will then be: Rule4=[2 6 19 58 14 43 49 67 39 38 35 26 7977 71 51 72 55 5 15 47 61 21 64 30 9 27]. This rule is not complete. Therule can not be applied in ‘transpose from’ mode without firstgenerating a complete 79 symbols originating sequence. However in the‘transpose to’ mode this rule can be used to hide symbols in anothersequence in a recoverable way.

The highest number in Rule4 is 79. Assume, as before, that the startingposition of a transposition is 1. Assume also that a sequence [1 2 3 4 56 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27] is goingto be ‘hidden’ in a sequence of all 0s. Clearly this is not reallyhiding the sequence. However for illustrative purposes the symbols to be‘hidden’ need to stand out. So a sequence of 27 different symbols isgoing to be transposed (or is going to overwrite or replace) 27 0s in asequence of 79 0s. The sequence with the second sequence transposed intoit will be: [0 1 0 0 19 2 0 0 26 0 0 0 0 5 20 0 0 0 3 0 23 0 0 0 0 12 270 0 25 0 0 0 0 11 0 0 10 9 0 0 0 6 0 0 0 21 0 7 0 16 0 0 0 18 0 0 4 0 022 0 0 24 0 0 8 0 0 0 15 17 0 0 0 0 14 0 13]. By inverting the Rule4 theoriginal sequence can be recovered.

Another ‘hiding’ scheme is provided as an illustrative example. Thefollowing ternary sequence Original=[0 0 0 1 0 1 1 1 2 1 0 2 0 0 2 1 2 02 2 2 0 1 2 2 1 1] with 27 elements will be hidden in a ternary sequenceLong79=[0 1 0 1 1 1 2 1 0 2 0 0 2 1 2 0 2 2 2 0 1 2 2 1 1 0 0 0 1 0 2 22 0 2 0 0 2 1 1 1 0 1 1 2 1 2 0 1 2 2 1 0 0 0 2 0 0 1 0 1 2 1 2 0 1 1 12 2 2 0 2 2 1 1 0 2 1] with 79 elements. By applying Rule4 in ‘transposeto’ mode the sequence Original is transposed into Long79 with resultingsequence Hide=[0 0 1 2 0 2 1 1 2 0 0 2 0 2 0 2 2 0 0 1 2 2 1 1 2 1 0 1 22 2 2 0 0 0 0 1 2 1 1 0 1 1 2 1 2 0 1 2 1 1 0 0 0 2 0 1 1 0 0 2 1 2 0 11 1 2 2 2 2 2 2 1 1 0 2 0]. By applying the inverse of Rule4 in‘transpose to’ mode one can recover Original from sequence Hide.

The effect of hiding a smaller sequence by transposition in a larger oneis shown in FIG. 15. Herein the combined auto-correlation of thesequence Long79 in thick line and the cross-correlation of Long79 withsequence Hide in thin line are shown. The effect of the Originalsequence on the overall cross-correlation is minimal. The peak of thecross-correlation coincides with the peak of the auto-correlation. Itspeak is about at 65, and is lower than the 79 of the auto-correlation.But detection by correlating Hide with Long79 is still fairly simple.One can do of course additional transpositions on the sequence Hide.

One can hide even one element in a multi-element sequence.

The hiding technique was demonstrated in an illustrative example usingternary sequences. It should be clear that the method, being anotheraspect of the present invention, can be applied using any n-valuedsequence to generate the hiding rule. Also the hiding sequence and thesequence to be hidden can be any n-valued sequence. Clearly the lengthof the sequences and the statistical make-up of the sequences and thehiding rule will influence how well a sequence can be hidden.

Using 4-valued Sequences

For illustrative purposes it will be shown that 4-valued sequences canalso be applied to create orthogonal transposition rules. Using 4-valued‘word’ methods one can create orthogonal transposition rules ofdifferent length. One can for instance create m-sequences of length(4^(p)−1). Herein p is the length of the applied words (or the length ofan LFSR 4-valued shift register. Some of the sequences will have thedesirable 2-level auto-correlation graph. For the illustrative example arule created by 2 4-valued element words will be used.

One sequence thus generated is the 16 elements sequence [2 0 0 1 1 0 3 32 3 0 2 1 3 1 2]. One can derive a transposition rule from this sequenceby first putting a copy of the first element of the sequence (2) at theend of the sequence and by considering each of 2 consecutive 4-valuedelements representing a decimal value. In 4-valued representation therule then is: [2 0; 0 0; 0 1; 1 1; 1 0; 0 3; 3 3; 3 2; 2 3; 3 0; 0 2; 21; 1 3; 3 1; 1 2; 2 2] or in decimal form: [8 0 1 5 4 3 15 14 11 12 2 97 13 6 10]. In order to make this a transposition rule working fromorigin 1 a 1 has to be added to all numbers thus creating: Rule42=[9 1 26 5 4 16 15 12 13 3 10 8 14 7 11].

Another 16 elements sequence is [0 0 1 0 2 2 1 1 2 0 3 3 1 3 2 3]. Anauto-correlation graph of this sequence is shown in FIG. 16. Using ruleRule42 in ‘transpose from’ mode on the sequence will create the sequence[2 0 0 2 2 0 3 2 3 1 1 0 1 3 1 3]. The auto-correlation of that sequenceis shown in FIG. 17. The cross-correlation of the original sequence withthe transposed sequence is shown in FIG. 18. It should be apparent toone of ordinary skills in the art that any reversible n-valuedpseudo-random sequence can be used to transpose and reverse a transposedsequence of symbols.

Other Applications of Transposition Rules

The transposition rules as developed in the present invention transposesymbols from one position in a sequence to another position whichsometimes can be the same position. It is another aspect of the presentinvention to interpret the generated rule of decimal numbers as actualpositions. The numbers represent individual slots or positions in aseries or frame of positions. A slot or a position may represent aspecific frequency band or a time slot. Each position or slot has aspecific number. While in a transposition one changes the position of asymbol, in this aspect of the invention a symbol is exchanged with whatwill be called a ‘user’. The ‘user’ is in essence a message or part of amessage that requires for instance a ‘time-slot’, a pulse, an assignedbandwidth or an assigned code to transmit the message or part of amessage. The transposition rule [a b c d] then has the followingmeaning: there are 4 users; each user will be assigned a transmissionresource (potentially for a finite time). There are 4 resources named‘a’, ‘b’, ‘c’ and ‘d’. User 1 is assigned resource ‘a’. User 2 isassigned resource ‘b’. User 3 is assigned resource ‘c’. User 4 isassigned resource ‘d’. In general one will assign a single resource (ora series of resources that may be considered a single resource) to asingle user. In order to prevent interference one will want to preventmultiple users having access to the same resource at the same time. Thisconcept is known as orthogonality. It is possible to assign moreresources to a single user. As long as orthogonality is observed havingaccess to more resources should not be a problem.

The following illustrative example will show how a transposition rule(in this case the 4-valued sequence based Rule42) can be applied in atime-hopping system. In a time-hopping system a transmission period isdivided in a frame with a discrete number of time slots. A user isrepresented by a sequence of pulses, wherein each user has at least onepulse in a time-slot in each timeframe. In general one wants each userto occupy a pulse in a different time slot in each consecutivetimeframe. Assuming that there are an equal number of users andtime-slots it should be clear that the assignment rule should beorthogonal (or non-conflicting).

Assume that there are 16 time slots and 16 users. The rule Rule42=[9 1 26 5 4 16 15 12 13 3 10 8 14 7 11] can then be applied to assigntime-slots. Because it may be required that the assigned time-slotsdiffer in each frame additional assignment rules are then required. Onecan use different assignment rules generated by for instance other 16element 4-valued sequences, generated by 2-element 4 valued words. Theadvantage is that unrelated sequences can be used. One can also derivethe next to be applied rule from the present rule. One way derive thenext assignment rule from the present is by shifting all elements oneposition to the right and move the last element to the first position.This will create Rule42sr1=[11 9 1 2 6 5 4 16 15 12 13 3 10 8 14 7]followed by Rule42sr2=[7 11 9 1 2 6 5 4 16 15 12 13 3 10 8 14]. One cancreate 16 different consecutive assignment rules. The results of thefirst 3 rules are shown in the pulse diagram with three consecutivetimeframes in FIG. 19. The x-axis shows three timeframes with 16time-slots. The y-axis shows the users. The thick short vertical linestell which time-slot is assigned to which user. Clearly this scheme isorthogonal. It repeats itself after 16 timeframes. Also the relationsbetween users and time-slots become predictable.

Another way to create orthogonal ‘hopping’ or placement rules is byadding a number ‘modulo-16+1’ to the elements of a previous rule, whichmakes each position shift by 1 in each next timeframe. The first 3timeframes as a result of this method are shown in FIG. 20. While thepatterns are orthogonal in each timeframe they are following a clearpattern. One can add also odd numbers to make the jumps seemingly lesspredictable.

Another way to create ‘n’ seemingly random ‘hopping’ patterns for ‘n’users is another aspect of the present invention. The method isexplained by using as an illustrative example based on a 16 elements4-valued sequence created by the 2 4-valued elements ‘word’ method. Thissequence, taking all consecutive 2 elements words and extending thesequence with a copy of the first element, can create a sequence of 16different decimal numbers. As there is no forbidden state among thewords the lowest decimal number in the sequence is 0 and the highest is15. To make the decimal numbers equivalent to positions each number isincreased by one. One can thus generate thousands and thousands ofdifferent decimal sequences. The next step is to use a generated rule tocreate 16 additional rules. This is done by assuming that a generatedsequence may be considered to be a sequence as well as a rule. So onecan actually ‘transpose’ the elements of a rule by the rule. Forillustrative purposes it is assumed that the rule executes the‘transpose from’ method.

A sequence of 16 different symbols can have 16 orthogonal sequencesconfigurations, as after 16 symbols a repeat of symbols has to takeplace. A successful sequence is Rule44=[7 10 8 15 12 14 6 5 3 11 9 4 1613 1 2]. Applying this rule first upon itself and then on its resultswill generate: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 rule 7 10 8 15 1214 6 5 3 11 9 4 16 13 1 2  1 6 11 5 1 4 13 14 12 8 9 3 15 2 16 7 10  214 9 12 7 15 16 13 4 5 3 8 1 10 2 6 11  3 13 3 4 6 1 2 16 15 12 8 5 7 1110 14 9  4 16 8 15 14 7 10 2 1 4 5 12 6 9 11 13 3  5 2 5 1 13 6 11 10 715 12 4 14 3 9 16 8  6 10 12 7 16 14 9 11 6 1 4 15 13 8 3 2 5  7 11 4 62 13 3 9 14 7 15 1 16 5 8 10 12  8 9 15 14 10 16 8 3 13 6 1 7 2 12 5 114  9 3 1 13 11 2 5 8 16 14 7 6 10 4 12 9 15 10 8 7 16 9 10 12 5 2 13 614 11 15 4 3 1 11 5 6 2 3 11 4 12 10 16 14 13 9 1 15 8 7 12 12 14 10 8 915 4 11 2 13 16 3 7 1 5 6 13 4 13 11 5 3 1 15 9 10 16 2 8 6 7 12 14 1415 16 9 12 8 7 1 3 11 2 10 5 14 6 4 13 15 1 2 3 4 5 6 7 8 9 10 11 12 1314 15 16 16 7 10 8 15 12 14 6 5 3 11 9 4 16 13 1 2 17 6 11 5 1 4 13 1412 8 9 3 15 2 16 7 10

By applying the transposing rule on its result one then achieves theresult as shown in the table. After 16 transpositions, i.e. from the17th transposition forward, the results will repeat. This means that 16orthogonal positions are available in a frame of 16 positions, and thatalso 16 superframes of 16 individual frames generated by a repeatedtransposition rule upon itself, wherein the transposition rule isdetermined from a pseudo-random 4-valued sequence, determined from2-symbol state words, are orthogonal.

One can then create 16 superframes, each superframe having 16 frames andeach frame having 16 positions, wherein a user or a channel isorthogonal on any other user for the duration of the 16 superframes,when all the superframes described by the table are synchronized.

A position in a frame can signify a position in time or a frequency.

It is possible that frames are not synchronized, in the sense thatframes may start at the same moment, but that each superframe can startat a any frame compared to other superframes. A measure of overlap canbe to combine 2 identical superframes of 16 frames into 32 frames andcheck the number of frames that a shifted superframe can have in commonwith this combined superframe. In the present example the highestpossible number of frames in common that a shifted superframe can havein common with part of two combined identical superframes is 4. One cancheck this with a computer program. This means that at least 12 framesare different. This is a significant enough difference to achievediscrimination between superframes, even if the superframes are notsynchronized.

One can actually find a better performing rule under this example:rule=[12 14 6 5 3 11 9 4 16 13 1 2 7 10 8 15]. This is a shifted versionof the earlier rule. Unsynchronized superframes under this rule haveonly 3 frames in common.

Another example of the method of creating superframes of hopping ortransposition rules is provided for a ternary case. A 3-element ternaryLFSR can generate the following ternary pseudo-random sequence of 26ternary elements: out3=[2 1 1 1 2 0 0 1 1 0 1 0 2 1 2 2 2 1 0 0 2 2 0 20 1]. By extending the sequence by the two first symbols one can createfrom overlapping words of 3 symbols the decimal sequence: [22 13 14 1518 1 4 12 10 3 11 7 23 17 26 25 21 9 2 8 24 20 6 19 5 16]. One can applythis rule upon itself for 26 times and create 26 superframes of 26frames. By shifting the rule, by moving the last element in the firstposition one can create a new rule. This is identical to starting theLFSR with another initial state. By using a computer program, ormanually if one so desires, one can then select the rule that creates 26different and orthogonal superframes. The result is shown in thefollowing table: hopping rule based on 3-element ternary pseudo-randomsequence 26 22 17 16 11 21 12 10 3 23 20 5 4 18 19 1 13 9 7 24 8 25 15 614 2 24 21 7 11 1 25 4 18 19 9 26 16 5 23 17 20 12 15 13 2 14 8 3 22 106 2 25 13 1 20 8 5 23 17 15 24 11 16 9 7 26 4 3 12 6 10 14 19 21 18 22 68 12 20 26 14 16 9 7 3 2 1 11 15 13 24 5 19 4 22 18 10 17 25 23 21 22 144 26 24 10 11 15 13 19 6 20 1 3 12 2 16 17 5 21 23 18 7 8 9 25 21 10 524 2 18 1 3 12 17 22 26 20 19 4 6 11 7 16 25 9 23 13 14 15 8 25 18 16 26 23 20 19 4 7 21 24 26 17 5 22 1 13 11 8 15 9 12 10 3 14 8 23 11 6 22 926 17 5 13 25 2 24 7 16 21 20 12 1 14 3 15 4 18 19 10 14 9 1 22 21 15 247 16 12 8 6 2 13 11 25 26 4 20 10 19 3 5 23 17 18 10 15 20 21 25 3 2 1311 4 14 22 6 12 1 8 24 5 26 18 17 19 16 9 7 23 18 3 26 25 8 19 6 12 1 510 21 22 4 20 14 2 16 24 23 7 17 11 15 13 9 23 19 24 8 14 17 22 4 20 1618 25 21 5 26 10 6 11 2 9 13 7 1 3 12 15 9 17 2 14 10 7 21 5 26 11 23 825 16 24 18 22 1 6 15 12 13 20 19 4 3 15 7 6 10 18 13 25 16 24 1 9 14 811 2 23 21 20 22 3 4 12 26 17 5 19 3 13 22 18 23 12 8 11 2 20 15 10 14 16 9 25 26 21 19 5 4 24 7 16 17 19 12 21 23 9 4 14 1 6 26 3 18 10 20 2215 8 24 25 17 16 5 2 13 11 7 17 4 25 9 15 5 10 20 22 24 19 23 18 26 21 314 2 8 7 11 16 6 12 1 13 7 5 8 15 3 16 18 26 21 2 17 9 23 24 25 19 10 614 13 1 11 22 4 20 12 13 16 14 3 19 11 23 24 25 6 7 15 9 2 8 17 18 22 1012 20 1 21 5 26 4 12 11 10 19 17 1 9 2 8 22 13 3 15 6 14 7 23 21 18 4 2620 25 16 24 5 4 1 18 17 7 20 15 6 14 21 12 19 3 22 10 13 9 25 23 5 24 268 11 2 16 5 20 23 7 13 26 3 22 10 25 4 17 19 21 18 12 15 8 9 16 2 24 141 6 11 16 26 9 13 12 24 19 21 18 8 5 7 17 25 23 4 3 14 15 11 6 2 10 2022 1 11 24 15 12 4 2 17 25 23 14 16 13 7 8 9 5 19 10 3 1 22 6 18 26 2120 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2620 6 19 5 16 22 13 14 15 18 1 4 12 10 3 11 7 23 17 26 25 21 9 2 8 24

The unsynchronized superframes of this rule have a maximum of 4 framesin common when one compares a shifted superframe with 2 combinedsuperframes. This means that without errors each superframes has atleast 22 unique or orthogonal frames. After a 4-valued and a ternaryexample it should be clear that the method for creating hopping ortransposition rules for superframes can be used with n-valued symbolsequences also.

According to another aspect of the present invention one can use thehopping or transposition rules to generate sequences of pulse trains,which will be used in a completely unsynchronized manner. This meansthat each user or channel may send a pulse train completely at randomtimes. The rules are used in the following manner, using the 16 framerule as an example. Each pulse train starts with a pulse and is followedby a number of time elements without any signal until a new pulseoccurs. The distance or time period between the pulses is determined bya number in the transposition or hopping rule. The 16-valued rulerule=[12 14 6 5 3 11 9 4 16 13 1 2 7 10 8 15] can be used as: a trainstarts with a pulse and is followed by 12 time elements of no signalwith a pulse after the 12^(th) time period when the previous pulseoccurred. This is followed by 14 time periods of no signal and again apulse occurs, etc. One can end the train with a pulse. One may alsostart the pulse train with a pulse. For instance one can apply the ruleas: start a train with a pulse and consider the rule “number” as theposition for the next pulse to occur following the present pulse. A rule“number” 1 then signifies the next position immediately following thepresent pulse. A pulse train is closed by a pulse as each pulse trainaccording to the rules occupies exactly 137 time periods. Shift andmatching of each of the pulse trains with a combined pulse train of twoidentical pulse trains will find 17 matching pulses if two pulse trainsare identical and maximal 8 when they are not matching. One can improvethe numbers by inversing pulses between +1 and −1 and thus reduce thematching pulses between shifted pulse trains to 6 under these rules.

Gold Sequences

In a co-pending U.S. application by the inventor entitled:SELF-SYNCHRONIZING GENERATION AND DETECTION OF SEQUENCES NOT USING LFSRSwhich is incorporated herein in its entirety, it is shown that Goldsequences, formed by combining pseudo random n-valued sequences formedfrom k n-valued symbol words can be used to from unique sequences ofoverlapping words of 2 k n-valued symbols.

As an illustrative example of the method a set of ternary Gold sequenceswill be used. The following ternary m-sequence of length 80 can begenerated by a ternary 4-element LFSR generator: [0 0 2 0 2 0 1 0 1 2 10 1 2 0 1 1 1 0 1 2 0 1 1 1 2 2 2 1 2 2 0 1 2 1 1 1 0 2 2 0 0 2 1 2 0 01 1 2 1 2 1 0 1 0 0 1 0 2 1 0 0 0 0 2 2 2 0 2 2 1 0 2 0 0 0 1 2 2 1 1 20 2 1 1]. A set of Gold sequences is formed by cyclically shifting andcombining with the following 80 symbol ternary m-sequence: [1 2 1 0 1 10 0 1 0 0 0 1 2 1 2 2 0 1 1 1 2 2 2 2 0 2 0 2 1 1 2 0 1 0 2 1 0 0 2 2 12 0 2 2 0 0 2 0 0 0 2 1 2 1 1 0 2 2 2 1 1 1 1 0 1 0 1 2 2 1 0 2 0 1 2 00 1]. One off the 80 generated ternary Gold sequences is: Gold1=[1 2 2 02 1 2 0 0 2 0 2 2 2 0 1 1 2 2 2 2 1 0 0 2 2 0 2 1 0 1 0 1 1 0 0 0 1 0 21 0 0 2 0 1 0 2 2 0 2 0 0 0 2 1 1 0 0 0 0 1 2 2 0 0 2 0 1 2 1 2 1 1 2 22 1 2 0]. One can use a computer program to take the first 4 digits ofthe sequence, determine the decimal value plus 1 of this ternary word,move one digit to the right, determine the next 4 digit word's decimalvalue plus 1, until one reaches the end of the sequence. This can bedone 77 times and can be translated into the following decimal sequence:[52 75 62 24 70 46 57 7 21 63 27 79 74 59 15 45 54 81 80 76 64 30 9 2575 62 22 65 31 11 32 13 37 28 24 12 35 22 64 30 7 20 58 12 36 25 75 6119 55 3 8 23 67 37 28 1 26 18 52 73 57 7 20 60 17 51 71 50 69 45 54 8078 70]. One can see that some words (for instance 2, 7 and 75) are usedmore than once.

One way to achieve a series of unique words in this Gold sequence is bycreating words of more than 4 symbols. It can be seen that re-occurringpatterns have a maximum length of 7 symbols so that words of length 8should be unique and enables the creation of a set of Gold sequences ofwhich each can be detected by using an addressable memory method. Theabove decimal sequence can be expressed in a decimal sequence of 73numbers formed by 8 symbol words: Gold1₁₃ 8=[4201 6040 4998 1870 56103708 4563 565 1694 5081 2121 6363 5967 4779 1214 3640 4357 6510 64086100 5178 2411 670 2009 6025 4952 1733 5197 2467 838 2513 976 2928 2222103 307 921 2761 1721 5161 2361 522 1564 4692 952 2854 1999 5997 48681481 4441 199 595 1783 5348 2922 2205 52 154 462 1384 4151 5892 4553 5371610 4829 1365 4095 5724 4049 5586 3634]. This sequence consists of 73unique 8 symbol ternary words.

One can take another sequence from this set of ternary Gold sequencesGold2=[0 1 1 0 1 1 1 1 1 1 0 1 1 0 0 2 1 1 1 1 1 1 1 0 1 1 2 2 2 2 2 2 10 0 1 0 2 1 0 2 0 2 0 2 1 1 2 1 1 1 1 1 1 2 2 0 1 1 1 1 1 1 1 2 1 1 0 00 0 0 0 1 2 2 1 2 0 1 2 0 2 0 2]. The translation of this sequence into8 symbol decimal words provides: Gold2_(—)8=[1013 3038 2552 1093 32773270 3248 3182 2984 2390 608 1823 5467 3278 3272 3255 3204 3051 25921215 3645 4373 6556 6544 6509 6403 6087 5138 2290 309 925 2775 1762 52862735 1643 4929 1664 4991 1850 5549 3524 4010 5469 3285 3292 3314 33803578 4172 5954 4739 1095 3284 3290 3307 3358 3511 3970 5347 2917 2189 618 53 159 475 1424 4272 6253 5637 3787 4800]. The sequence Gold1₁₃ 8 issignificantly different from Gold2_(—)8. All Gold sequences of the setwill generate significantly different 8 symbol word sequences.

All Gold sequences of the set generated by two different 4-elementternary LFSRs will generate different 8 symbol word sequences. Not onlydo words not repeat within a sequence, they will also not repeat withinthe set of sequences. Consequently decimal numbers based on these wordsare unique to a sequence of a set. This rule has also been tested on forinstance binary Gold sequences, wherein a set of binary Gold-sequenceswas generated by two 6-elements LFSRs. One can then describe eachsequence of that set by 12 bits overlapping words. Each word (and itsdecimal equivalent) is unique to a sequence and will only appear once ina set of sequences.

Accordingly a set of Gold sequences can be modified in such a may that,assuming that a Gold sequence was formed from 2 pseudo-random sequencesbased on words of k n-valued symbols, then each Gold sequence from theset can be expressed as a unique set of decimal number derived fromoverlapping words of 2 k n-valued symbols. It is not required to use allsequences of a set. One can renumber or normalize the to be usedsequences of decimal numbers by substituting the lowest number with 1,the next lowest number with 2 etc. Thus each Gold sequence thenrepresents a certain number of frames, each frame having a decimalnumber that is unique to a set of Gold sequences.

It should be clear that each Gold sequence of p symbols made fromsequences that can be decomposed in p words of k symbols, can bedecomposed into p words of 2*k symbols.

Each hopping rule, derived from a Gold sequence in a set is orthogonalto another rule from the set. When a Gold sequence has p symbols, thereexist a maximum of p different Gold sequences in the set. Theauto-correlation of a Gold sequence has a peak value corresponding withthe number of symbols. A Gold sequence is different from another Goldsequence when the maximum correlation between the two sequences has nopeak like the one that occurs in the auto-correlation. A Gold sequenceis different from a shifted version of itself and orthogonal is mostcases, and can be used as such. However shifted Gold sequences requiresynchronization, because if shifted over the full p symbols the twoshifted sequences will be coinciding.

Accordingly a Gold sequence from a set represented in its decimalhopping rule form is always orthogonal to any other sequence in the setof Gold sequences represented as a decimal hopping rule.

In general the term pseudo-random sequence is used for a class ofsequences generated by LFSRs. LFSRs have a forbidden state andaccordingly pseudo-random sequences generated by LFSRs do not comprise aforbidden state. The inventor has demonstrated a ‘word’ method whereinone can generate sequences comprising all possible words. Accordinglypseudo random sequences in aspects of the present invention comprisesequences with and without forbidden words, or of length n^(k)−1n-valued symbols when generated by LFSR with k elements or of lengthn^(k) n-valued symbols when generated by k n-valued word methods.

The general purpose of the present invention is to provide novel methodsand systems which can be applied in the encipherment of sequences ofsymbols by transposition, using digital sequences with pseudo-noise orpseudo-noise like properties and the decipherment of the encryptedsequences. Sequences are made of series of symbols with an assignedposition relative to an assumed or assigned origin or anchor point. Theindividual symbols and their order in a sequence may represent anelectrical or optical signal. The position of a symbol may represent aphysical order, a time slot, a frequency, a color or any otherphenomenon or concept that can be represented as a position.

It is one aspect of the present invention to provide a system that canexecute the transposition rules. One such system is provided as adiagram 2100 in FIG. 21. The system has a module 2101 that eitherreceives a sequence through input 2102 or generates a sequenceinternally and creates a transposition rule. A sequence of symbols isinputted on 2103 and is deserialized and stored in a module 2105.Transposition takes place by transferring the symbols from 2105 to amemory and serializing unit 2106. The order in which the symbols arestored in 2106 is determined by the rule as executed by 2101. In thisexample in a first embodiment the execution of the transposition iscontrolled by gates that control the transfer of symbols from 2105 to2106. The gates (of which one, 2107, is identified in the diagram) arecontrolled by the transposition rule in 2101. After the symbols aretransferred from 2105 to 2106 the transposed sequence is outputted on2104. The circuit is controlled by a clock signal 2108. The system 2100also has an internal clock circuit applying the external clock to countthe number of symbols to be transposed and to initiate the outputting ofthe transposed sequence. Transposing inherently works on fixed lengthsequences, and processing delay will occur. One may diminish delays byfor instance having one sequence buffered while another sequence isbeing processed. FIG. 22 shows a diagram of a system reversing thetransposition. The system for reversal is essentially a mirror image ofFIG. 21. A significant difference is that module 2201 creates thereversing rule of the module 2101.

FIG. 23 shows a diagram of a system that is an aspect of the presentinvention that creates and executes a plurality of transposition orhopping rules in a module 2301 in accordance with another aspect of thepresent invention, which is here illustrated for a frequency hoppingsystem. The system 2300 is provided (for illustrative purposes) with 4input signals, 2302, 2303, 2304 and 2305, for example all in base-bandfrequencies. The purpose of the system is to create one signal in thetime-domain that has the 4 input signals in 4 different frequency bands.The system has a set of variable modulators 2306, of which the settingsare controlled by a hopping rule 2301. During a finite time the settingsof the modulators remain unchanged after which they are changed in anorthogonal fashion and in accordance with the methods of the presentinvention into a new setting. Because the selected modulationfrequencies are orthogonal (and assuming that sufficient channelseparation is achieved) the four outputted signals 2312, 2313, 2314 and2315 will not interfere with each other and can be combined by a module2308 into a single fdm signal 2307.

The transposition and hopping rules can be created and executed using aprocessor with memory. Such a processor may be part of a computer;however it may also be dedicated logic in customized processingcircuits. Multi-valued signals and symbols may be processed by binarylogic wherein each symbol is represented as a binary word. Words ofn-valued symbols may be represented as a plurality of binary words. Onemay also apply logic circuitry that is able to implement multi-valuedlogic switching functions. It is to be understood that the invention isnot limited in its application to the details of construction and to thearrangements of the components set forth in the description orillustrated in the drawings. The invention is capable of otherembodiments and of being practiced and carried out in various ways.Also, it is to be understood that the phraseology and terminologyemployed herein are for the purpose of the description and should not beregarded as limiting.

The following patent applications, including the specifications, claimsand drawings, are hereby incorporated by reference herein, as if theywere fully set forth herein: (1) U.S. Non-Provisional Patent ApplicationNo. 10/935,960, filed on Sep. 8, 2004, entitled TERNARY AND MULTI-VALUEDIGITAL SCRAMBLERS, DESCRAMBLERS AND SEQUENCE GENERATORS; (2) U.S.Non-Provisional Patent Application No. 10/936,181, filed Sep. 8, 2004,entitled TERNARY AND HIGHER MULTI-VALUE SCRAMBLERS/DESCRAMBLERS; (3)U.S. Non-Provisional Patent Application No. 10/912,954, filed Aug. 6,2004, entitled TERNARY AND HIGHER MULTI-VALUE SCRAMBLERS/DESCRAMBLERS;(4) U.S. Non-Provisional Patent Application No. 11/042,645 , filed Jan.25, 2005, entitled MULTI-VALUED SCRAMBLING AND DESCRAMBLING OF DIGITALDATA ON OPTICAL DISKS AND OTHER STORAGE MEDIA; (5) U.S. Non-ProvisionalPatent Application No. 11/000,218, filed Nov. 30, 2004, entitled SINGLEAND COMPOSITE BINARY AND MULTI-VALUED LOGIC FUNCTIONS FROM GATES ANDINVERTERS; (6) U.S. Non-Provisional Patent Application No. 11/065,836filed Feb. 25, 2005, entitled GENERATION AND DETECTION OF NON-BINARYDIGITAL SEQUENCES; (7) U.S. Non-Provisional Patent Application No.11/139,835 filed May 27, 2005, entitled MULTI-VALUED DIGITAL INFORMATIONRETAINING ELEMENTS AND MEMORY DEVICES; (8) U.S. Provisional PatentApplication No. 60/695,317 filed on Jun. 30, 2005 entitled THE CREATIONAND DETECTION OF BINARY AND NON_BINARY PSEUDO-NOISE SEQUENCES NOT USINGLFSR CIRCUITS; and (9) U.S. patent application Ser. No. 11/427,498 filedon Jun. 29, 2006 entitled THE CREATION AND DETECTION OF BINARY ANDNON-BINARY PSEUDO-NOISE SEQUENCES NOT USING LFSR CIRCUITS.

1. A method for transposing symbols in a first sequence into a secondsequence, by applying a reversible transposition rule derived from athird sequence of n-valued symbols with n≧2 which is pseudo-random. 2.The method as claimed in claim 1, wherein the third sequence is createdfrom n^(k)−1 of n^(k) different words of k n-valued symbols and hasn^(k)−1 n-valued symbols and wherein one of the n^(k) words is aforbidden word.
 3. The method as claimed in claim 1, wherein the thirdpseudo-random sequence is created from n^(k) different words of kn-valued symbols and has n^(k) symbols.
 4. The method as claimed inclaim 2, wherein the reversible transposition rule comprises:decomposing the third sequence into consecutive 2^(k)−1 words ofn-valued symbols; replacing each consecutive 2^(k)−1 word by a numberrepresenting a decimal value of each word to form a sequence of decimalnumbers; replacing the decimal number 0 in the sequence of decimalnumbers by a decimal value of the forbidden word if the word representedby decimal 0 is not the forbidden word of the third sequence; and addinga 1 to all numbers in the sequence of decimal numbers if the wordrepresented by 0 is the forbidden word of the third sequence.
 5. Themethod as claimed in claim 4, wherein the reversible transposition rulecomprises: placing a symbol in the p^(th) position in the secondsequence that is identical to a symbol at a position in the firstsequence, the position in the first sequence being equivalent to thevalue of the number in the p^(th) position in the sequence of decimalnumbers.
 6. The method as claimed in claim 4, wherein the reversibletransposition rule comprises: placing a symbol in p^(th) position in thefirst sequence in a position in a second sequence, the position in thesecond sequence being equivalent to the value of the number in thep^(th) position in the sequence of decimal numbers.
 7. The method asclaimed in claim 3, wherein the reversible transposition rule comprises:decomposing the third sequence into consecutive n^(k) words of n-valuedsymbols; replacing each word by a number representing a decimal value ofeach word thus forming a sequence of decimal numbers; and adding a 1 toall numbers in the sequence of decimal numbers.
 8. A method for creatinga plurality of orthogonal hopping rules from a first pseudo-randomsequence of n-valued symbols, comprising the steps: 1) creating atransposition rule as a sequence of decimal numbers from the firstpseudo-random sequence; 2) making the sequence of decimal numbers asecond sequence and transposing the second sequence into a thirdsequence according to the transposition rule; 3) creating a next thirdsequence by applying the transposition rule to the third sequence; 4)making the next third sequence the third sequence; and 5) repeatingsteps 3) and 4) till one of the next third sequences is identical to apreviously created sequence.
 9. A method for creating a plurality oforthogonal hopping rules from a set of p Gold sequences, each Goldsequence having p n-valued symbols and each Gold sequence can be formedfrom a first and second sequence, the first and second sequences beingable to be decomposed into p different words of k n-valued symbols,comprising: selecting a Gold sequence from the set; decomposing theselected Gold sequence from the set into a first sequence of p differentwords of 2*k n-valued symbols; replacing each word of 2*k n-valuedsymbols in the first sequence by a number representing a decimal valueof the word to form a sequence of decimal numbers as a hopping rule;selecting a new Gold sequence from the set; and repeating the previoussteps for no more than p times.
 10. The method as claimed in claim 9,further comprising normalizing the hopping rules to the total number ofhops included in the hopping rules.
 11. A system for transposing symbolsin a first sequence into a second sequence by applying a reversibletransposition rule, comprising: an input providing the first sequence ofsymbols; an output providing the second sequence of transposed symbols;and a module receiving the input and providing the output, the moduleimplementing a reversible transposition rule derived from a thirdsequence of n-valued symbols with n≧2 which is pseudo-random.
 12. Thesystem as claimed in claim 11, wherein the third sequence is createdfrom n^(k)−1 of n^(k) different words of k n-valued symbols and hasn^(k)−1 n-valued symbols and wherein one of the n^(k) words is aforbidden word.
 13. The system as claimed in claim 11, wherein thepseudo-random sequence is created from n^(k) different words of kn-valued symbols and has n^(k) symbols.
 14. The system as claimed inclaim 12, wherein the transposition rule is created by: decomposing thethird sequence into consecutive 2^(k)−1 words of n-valued symbols;replacing each word by a number representing a decimal value of eachword to form a sequence of decimal numbers; replacing a decimal number 0in the sequence of decimal numbers by the decimal value of the forbiddenword if a word represented by decimal 0 is not the forbidden word of thethird sequence; and adding a 1 to all numbers in the sequence of decimalnumbers if the word represented by 0 is the forbidden word of the thirdsequence.
 15. The system as claimed in claim 14, wherein thetransposition rule comprises the steps: placing a symbol in the p^(th)position in a second sequence that is identical to a symbol in theposition in the first sequence, the position in the first sequence beingequivalent to the value of the number in the p^(th) position in thesequence of decimal numbers.
 16. The system as claimed in claim 14,wherein the transposition rule comprises the steps: placing a symbol inp^(th) position in a first sequence in a position in a second sequence,the position in the second sequence being equivalent to the value of thenumber in the p^(th) position in the sequence of decimal numbers. 17.The system as claimed in claim 13, wherein the reversible transpositionrule comprises: decomposing the third sequence into consecutive n^(k)words of n-valued symbols; replacing each word by a number representinga decimal value of each word to form a sequence of decimal numbers; andadding a 1 to all numbers in the sequence of decimal numbers.